My tredmill works but after few mins I get burnt smell?

limited info generic answer seem to have a few things wrong ---driving through [puddle] may have or did damage electrical system like alternator which is not charging battery causing stall no start -but will need to test the battery alternator and electrical system ---also operating without oil water could also have gotten into engine and pretty much damage /destroy it need to inspect and check internals of engine

Slipping transmission

Does the smell happen if you are running the hvac on just vent ?
The ac itself would not cause a smell unless the clutch on the compressor were slipping and you would probably hear that.
Its a sealed system just like your fridge at home.
So it could be something in the engine compartment that is coming thru the vent when the ac is on. The most common smell is a musty or mold smell from moisture inside the hvac.

usually ,when using the egg discription it referred to as a rotten egg smell ,not burnt.. if this is the case you may have a catalytic converter that is in need of replacement..

smell radio sound crazy you will smell an oder of burnt wireing if not radio still good check wiring and ground and follow remote wiring see if has reset button on radio

burnt or unburnt gas? if burnt gas smell look for bad manifold or manifold to pipe (donut) gasket.
if unburnt smell look for gas line leaks

Check the compressor when you notice the smell. See if its running or not. It could also be the blower motor itself. I'd suggest checking the blower motor first off. Those usually only last a few years, and if it got wet some how it could have rusted or corroded on the inside. Or a wire broke on the inside and it shorts out to ground which could explain the smell.

It sounds like the bands inside the transmission are burnt. one way to check is pull you trans dipstick and smell the fluid and see if it smells burnt and the fluid should also be bright red. i have burnt a few trans myself and they did that.lol

If you jack up car, can you rotate each wheel and see each halfshaft turning all the way into the transmissin?

relay was stuck.  Now you replace the relay and a few burnt fuses and the burnt wiring.  You should have fixed this at the first warning sign of a problem...sadly, now you pay the price for not doing so.